Answer

Correct Option: (i) $m \le 2^{n}$

Explanation

NFA → DFA subset construction:

• Each DFA state corresponds to a subset of NFA states.
• With $n$ NFA states, there are at most $2^{n}$ subsets.
• Hence the DFA obtained (before minimization) has $\le 2^{n}$ states.
• Minimized DFA $M$ has $m \le$ number of subset-construction DFA states $\le 2^{n}$.

So (i) is always true.

Other options:

• (ii) $n \le m$ is false: NFA can be exponentially smaller (e.g., language of $n^{th}$ from end needs $n+1$ NFA states but $2^{n}$ DFA states → $n < m$ sometimes, but not necessarily; sometimes DFA is smaller).
• The converse $m \ge n$ is not guaranteed.

Final Answer

$m \le 2^{n}$ → Option (i). (Minimized DFA minimal, so $m$ is even tighter, but certainly $\le 2^{n}$.)